Fourier type operators on Orlicz spaces and the role of Orlicz Lebesgue exponents

TL;DR

This paper extends classical Fourier analysis results to Orlicz spaces, establishing operator continuity and wave-front properties, and linking Orlicz space functions to Lebesgue exponents, broadening the scope of Fourier multiplier theorems.

Contribution

It generalizes H{"o}rmander's Fourier multiplier theorem to Orlicz spaces and explores the connection between Orlicz functions and Lebesgue exponents.

Findings

Extended Fourier multiplier theorems to Orlicz spaces.

Established continuity and wave-front properties for Fourier operators on Orlicz spaces.

Linked Orlicz functions to Lebesgue exponents $p_\Phi$ and $q_\Phi$.

Abstract

We deduce continuity and (global) wave-front properties of classes of Fourier multipliers, pseudo-differential, and Fourier integral operators when acting on Orlicz spaces, or more generally, on Orlicz-Sobolev type spaces. In particular, we extend H{\"o}rmander's improvement of Mihlin's Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions $\Phi$ of the Orlicz spaces are linked to properties of certain Lebesgue exponents $p_\Phi$ and $q_\Phi$ emerged from $\Phi$.